This ebook contains chosen contributions by means of academics on the 3rd annual Formation d’Automatique de Paris. It presents a well-integrated synthesis of the most recent pondering in nonlinear optimum keep watch over, observer layout, balance research and structural houses of linear platforms, with no the necessity for an exhaustive literature overview. The across the world identified individuals to this quantity symbolize a number of the so much respected keep an eye on facilities in Europe.

The optimum keep watch over of versatile constructions is an lively sector of study. the most physique of labor during this region is anxious with the regulate of time-dependent displacements and stresses, and assumes linear elastic stipulations, specifically linear elastic fabric habit and small defor- tion. See, e. g. , [1]–[3], the collections of papers [4, 5], and references therein.

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0) and is the response of to a smooth control denoted uγ . The control can be normalized to zero by the feedback v = u − uγ . Then, diﬀerentiating as before H1 (z) = 0, one gets that, everywhere p(t), adk F0 · F1 (γ(t)) = 0, k ≥ 0. We proved the following. 4. Let z be a smooth singular extremal on [0, T ], corresponding to a singular control identiﬁed to zero. 12) everywhere on [0, T ]. 12), adk H0 · H1 denotes the k-th Poisson bracket of H1 with H0 . This is clearly equivalent to the following lemma.

21. The value function Fmax → T (Fmax ) mapping to each positive maximum thrust the corresponding minimum time is right continuous for the transfer problem (2D or 3D, constant mass or not). As a matter of fact, we will use a decreasing sequence of thrusts bounds k (Fmax )k . Therefore, right continuity of the value function is enough to guark antee that T (Fmax ) tends to T (Fmax ) when the thrusts decrease to Fmax . But while mere discrete homotopy is used to initialize the search for p0 , a much more precise guess for the minimum time is available.

The time tc is called conjugate if there exists a Jacobi ﬁeld vertical both at t = 0 and tc . In this case, x(tc ) is said to be conjugate to x(0) along the reference solution. 8. If z(t, t0 , z0 ) is the integral curve of H (t, z) with initial condition z0 at t = 0, the exponential mapping at t is deﬁned by expx0 ,t : p0 → Π(z(t, x0 , p0 )). The following result is a consequence of the previous analysis. 9. Let z be a reference extremal with initial condition z0 = (x0 , p0 ) deﬁned on [0, T ].